CSA member Missy Hartman shared this tip for washing and keeping greens for several weeks in the fridge.
I wash the greens in 3 changes of water to get all the dirt and sand out. Lay them on a big bath towel and roll up gently to remove excess moisture. Lay out a long piece of paper toweling and layer the greens on top. They don’t have to be completely dry, in fact you should have a little moisture still on the greens, just not so much that it will rot them. These should be pretty much a single layer, if you have more than that, put another piece of paper toweling on top. Roll these up and put in a gallon sized ziploc bag. They will keep for 2 weeks to a month. We ate some tat soi tonight that was bagged this way for a month.